Flights for May 2012 won't be available for quite some time yet. It's unlikely that there would be many last minute bargains, so you should probably book as soon as possible.
The nearest airport to Altinkum is Bodrum. According to the airport's website, the only direct flights to Bodrum from the UK are from Manchester (with First Choice) and Newcastle (with Thomas Cook). You might also find some charter flights, from other airports, operating to Bodrum but charter operators usually only offer seats on their aircraft when they know that they'll have spare capacity after selling their own inclusive holidays. (Also, charter flights usually only operate on one day per week, to fit in with the tours operators' holidays. They might not be on the days that you require).
Izmir Airport is probably the only realistic alternative. According to that airport's website, there are direct flights from Gatwick (BA, Thomas Cook & Thomson), Stansted (SunExpress), Belfast (Onur Air & Thomas Cook), Birmingham (Thomas Cook & Thomson), Bristol (Thomas Cook), East Midlands (Thomas Cook), Manchester (Thomas Cook & Thomson) and Newcastle (Thomas Cook). However it should be noted that most of those flights are seasonal; they might not commence early enough in May to meet your needs.
http://www.izmir-airport.info/airlines.html
Chris